Variety of Evidence

Abstract

Varied evidence confirms more strongly than less varied evidence, ceteris paribus. This epistemological Variety of Evidence Thesis enjoys widespread intuitive support. We put forward a novel explication of one notion of varied evidence and the Variety of Evidence Thesis within Bayesian models of scientific inference by appealing to measures of entropy. Our explication of the Variety of Evidence Thesis holds in many of our models which also pronounce on disconfirmatory and discordant evidence. We argue that our models pronounce rightly. Against a backdrop of failures of the Variety of Evidence Thesis, the intuitive case for the Variety of Evidence Thesis emerges strengthened. Our models do however not support the general case for the thesis since our explication of it fails to hold in certain cases. The parameter space of this failure is explored and an explanation for the failure is offered.

Notes

Acknowledgements

I would like to thank Stephan Hartmann, Barbara Osimani, Roland Poellinger and Christian Wallmann for very helpful comments and discussions. Thanks are also due to George Pólya for teaching me about reasoning by analogy and the value of limiting cases, both of which were most helpful for devising proofs. This work is supported by the European Research Council (Philosophy of Pharmacology: Safety, Statistical standards and Evidence Amalgamation, grant 639276).

Appendix 1: Proofs of Main Results

We now give the longer proofs. The propositions to be proved are re-stated for ease of reference.

Proposition 1

For bodies of evidence\({\mathcal {E}},{\mathcal {E}}'\)with\(|{\mathcal {E}}|=|{\mathcal {E}}'|\)it holds that

Fortunately, all those terms which do not contain \(\alpha _{00}\) nor \(\alpha _{11}\) (these are precisely those terms with \(P({\bar{c}}|{\bar{h}})P(c|{\bar{h}})P({\bar{c}}|{\bar{h}})P(c|{\bar{h}})\)) cancel out. Furthermore, all terms which contain \(\alpha _{00}^2\) and all terms containing \(\alpha _{11}^2\) cancel out.

For the terms containing \(\alpha _{00}\) and \(\alpha _{11}\) we find

Appendix 2: High Arity Variables

We now address the claim that the so-far established technical results, also hold for models which employ higher arity hypothesis and/or consequence variables.

Denote by \(h^2,h^3,\ldots \) the values of H different from h and by \(c^2,c^3,\ldots \) the values of consequence variable C different from c. \(h^0\) is the (possibly infinite) disjunction of the \(h^i\) with \(i\ge 2\). \(c^0\) is the (possibly infinite) disjunction of the \(c^k\) with \(k\ge 2\). To simplify notation we let \(P(c_j^0|h^i):=1-P(c_j|h^i)\).

The ceteris paribus conditions are that for every consequence variables \(C_j\) and every evidence variable E pertaining to the consequence variable \(C_j\) it holds that

This formalises the thought that all values of the hypothesis variable H different from h are equal. Furthermore, the conditional probability of the evidence does not depend on the particular value \(c^k\). We can hence define our Bayes factors in the usual way unequivocally as \(P(e|c)/P(e|{\bar{c}})\).

Proposition 5

If (20) and (21) hold, then Corollarys1and2hold for higher arities, too.

Proof

Using the ceteris paribus condition for the second and third equality, we find

Whenever the term ‘\(P(e|{\bar{c}}_j)\cdot P({\bar{c}}_j|{\bar{h}})\cdot P({\bar{h}})\)’ appears in a proof for the basic model it is replaced by the term ‘\(\sum _{i=2}\sum _{k=2}P(h^i)\cdot P(c^k_j|h^i)\cdot P(e|c^k_j)\)’. Since both terms are equal, no new difficulties arise.

2.

The terms of the form ‘\(P({\bar{c}}_j|{\bar{h}})\)’ for \(i\ge 2\) are replaced by terms of the form \(\sum _{k\ge 2}P(c_j^k|h^i)\)’. The latter is equal to \(P(c_j^0|h^i)\). By the first ceteris paribus assumption for greater arities, these terms are equal. \(\square \)

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